CDA6530: Performance Models of Computers and Networks. Chapter 2: Review of Practical Random Variables

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1 CDA6530: Performance Models of Computers and Networks Chapter 2: Review of Practical Random Variables

2 Two Classes of R.V. Discrete R.V. Bernoulli Binomial Geometric Poisson Continuous R.V. Uniform Exponential, Erlang Normal Closely related Exponential Geometric Normal Binomial, Poisson 2

3 Definition Random variable (R.V.) X: A function on sample space X: S! R Cumulative distribution function (CDF): Probability distribution function (PDF) Distribution function F X (x) = P(X x) Can be used for both continuous and discrete random variables 3

4 Probability density function (pdf): Used for continuous R.V. F X (x) = R x 1 f X (t)dt Probability mass function (pmf): Used for discrete R.V. Probability of the variable exactly equals to a value f X (x) = P (X = x) f X (x) = df X(x) dx 4

5 Bernoulli A trial/experiment, outcome is either success or failure. X=1 if success, X=0 if failure P(X=1)=p, P(X=0)=1-p Bernoulli Trials A series of independent repetition of Bernoulli trial. 5

6 Binomial A Bernoulli trials with n repetitions Binomial: X = No. of successes in n trails X» B(n, p) P (X = k) f(k; n; p) =! = n! (n k)!k! where à n k à n k! p k (1 p) n k 6

7 Binomial Example (1) A communication channel with (1-p) being the probability of successful transmission of a bit. Assume we design a code that can tolerate up to e bit errors with n bit word code. Q: Probability of successful word transmission? Model: sequence of bits trans. follows a Bernoulli Trails Assumption: each bit error or not is independent P(Q) = P(e or fewer errors in n trails) = P e i=0 f(i; n; p) = P e i=0 Ã n i! p i (1 p) n i 7

8 Binomial Example (2) ---- Packet switching versus circuit switching Packet switching allows more users to use network! 1 Mb/s link each user: 100 kb/s when active active 10% of time circuit-switching: 10 users packet switching: with 35 users, prob. of > 10 active less than.0004 N users 1 Mbps link Q: how did we know ? Introduction 1-8

9 Geometric Still about Bernoulli Trails, but from a different angle. X: No. of trials until the first success Y: No. of failures until the first success P(X=k) = (1-p) k-1 p P(Y=k)=(1-p) k p X Y 9

10 Poisson Limiting case for Binomial when: n is large and p is small n>20 and p<0.05 would be good approximation Reference: wiki =np is fixed, success rate X: No. of successes in a time interval (n time units) P (X = k) = e k k! Many natural systems have this distribution The number of phone calls at a call center per minute. The number of times a web server is accessed per minute. The number of mutations in a given stretch of DNA after a certain amount of radiation. 10

11 Continous R.V - Uniform X: is a uniform r.v. on (, ) if f(x) = 8 < 1 ; : if < x < 0 otherwise Uniform r.v. is the basis for simulation other distributions Introduce later 11

12 Exponential r.v. X: f(x) = F X (x)= 1-e - x 8 < : e x ; if x 0 0 if x < 0 Very important distribution Memoryless property Corresponding to geometric distr. 12

13 r.v. X (k-th Erlang): Erlang K-th Erlang is the sum of k Exponential distr. 13

14 Normal r.v. X: f(x) = 1 ¾ p 2¼ e (x ¹)2 =(2¾ 2) ; 1 < x < 1 Corresponding to Binomial and Poisson distributions 14

15 If X~N(¹, ¾ 2 ), then Normal r.v. Z=(X-¹)/¾ follows standard normal N(0,1) P(Z<x) is denoted as (x) (x) value can be obtained from standard normal distribution table (next slide) Used to calculate the distribution value of a normal random variable X~N(¹, ¾ 2 ) P(X< ) = P(Z < ( -¹)/¾) = ( ( -¹)/¾ ) 15

16 Standard Normal Distr. Table P(X<x) = (x) (-x) = 1- (x) why? About 68% of the area under the curve falls within 1 standard deviation of the mean. About 95% of the area under the curve falls within 2 standard deviations of the mean. About 99.7% of the area under the curve falls within 3 standard deviations of the mean. 16

17 Normal Distr. Example An average light bulb manufactured by Acme Corporation lasts 300 days, 68% of light bulbs lasts within 300+/- 50 days. Assuming that bulb life is normally distributed. Q1: What is the probability that an Acme light bulb will last at most 365 days? Q2: If we installed 100 new bulbs on a street exactly one year ago, how many bulbs still work now on average? What is the distribution of the number of remaining bulbs? Step 1: Modeling X~N(300, 50 2 ) ¹=300, ¾=50. Q1 is P(X 365) define Z = (X-300)/50, then Z is standard normal For Q2, # of remaining bulbs, Y, is a Bernoulli trial with 100 repetitions with small prob. p = [1- P(X 365)] Y follows Poisson distribution (approximated from Binomial distr.) E[Y] = λ= np = 100 * [1- P(X 365)] 17

18 Memoryless Property Memoryless for Geometric and Exponential Easy to understand for Geometric Each trial is independent how many trials before hit does not depend on how many times I have missed before. X: Geometric r.v., P X (k)=(1-p) k-1 p; Y: Y=X-n No. of trials given we failed first n times P Y (k) = P(Y=k X>n)=P(X=k+n X>n) = P (X=k+n;X>n) P (X>n) = (1 p)k+n 1 p (1 p) n = P (X=k+n) P (X>n) = p(1 p) k 1 = P X (k) 18

19 pdf: probability density function Continuous r.v. f X (x) pmf: probability mass function Discrete r.v. X: P X (x)=p(x=x) Also denoted as P X (x) or simply P(x) 19

20 Discrete r.v. X E[X] = kp X (k) Continous r.v. X E[X] = Mean (Expectation) Z 1 1 kf(k)dk Bernoulli: E[X] = 0(1-p) + 1 p = p Binomial: E[X]=np (intuitive meaning?) Geometric: E[X]=1/p (intuitive meaning?) Poisson: E[X]= (remember =np) 20

21 Mean Continuous r.v. Uniform: E[X]= ( + )/2 Exponential: E[X]= 1/ K-th Erlang E[X] = k/ Normal: E[X]=¹ 21

22 Function of Random Variables R.V. X, R.V. Y=g(X) Discrete r.v. X: E[g(X)] = g(x)p(x) Continuous r.v. X: E[g(X)] = Z 1 1 g(x)f(x)dx Variance: Var(X) = E[ (X-E[X]) 2 ] = E[X 2 ] (E[X]) 2 22

23 Joint Distributed Random Variables F XY (x,y)=p(x x, Y y) F XY (x,y)=f X (x)f Y (y) if X and Y are independent F X Y (x y) = F XY (x,y)/f Y (y) E[ X + Y]= E[X]+ E[Y] If X, Y independent E[g(X)h(Y)]=E[g(X)] E[h(Y)] Covariance Measure of how much two variables change together Cov(X,Y)=E[ (X-E[X])(Y-E[Y]) ] = E[XY] E[X]E[Y] If X and Y independent, Cov(X,Y)=0 23

24 Limit Theorems - Inequality Markov s Inequality r.v. X 0: 8 >0, P(X ) E[X]/ Chebyshev s Inequality r.v. X, E[X]=¹, Var(X)=¾ 2 8 k>0, P( X-¹ k) ¾ 2 /k 2 Provide bounds when only mean and variance known The bounds may be more conservative than derived bounds if we know the distribution 24

25 Inequality Examples If =2E[X], then P(X ) 0.5 A pool of articles from a publisher. Assume we know that the articles are on average 1000 characters long with a standard deviation of 200 characters. Q: what is the prob. a given article is between 600 and 1400 characters? Model: r.v. X: ¹=1000, ¾=200, k=400 in Chebyshev s P(Q) = 1- P( X-¹ k) 1- (¾/k) 2 =0.75 If we know X follows normal distr.: The bound will be tigher 75% chance of an article being between 760 and 1240 characters long 25

26 Strong Law of Large Number i.i.d. (independent and identically-distributed) X i : i.i.d. random variables, E[X i ]=¹ With probability 1, (X 1 +X 2 + +X n )/n ¹, as n1 Foundation for using large number of simulations to obtain average results 26

27 Central Limit Theorem X i : i.i.d. random variables, E[X i ]=¹ Var(X i )=¾ 2 X Y= 1 + X X n n¹ ¾ p n Then, Y» N(0,1) as n1 The reason for why normal distribution is everywhere Sample mean is also normal distributed Sample mean ¹X ¹X = nx i=1 X i =n E[ ¹X] = ¹ V ar( ¹X) = ¾ 2 =n What does this mean? 27

28 Example Let X i, i=1,2,, 10 be i.i.d., X i is uniform X10 distr. (0,1). Calculate E[X i ]=0.5, Var(X i )=1/12 P ( 10 X i=1 P ( i=1 X i > 7) P 10 X X i > 7) = P ( q i=1 i 5 10(1=12) > 7 5 q10(1=12) ) ¼ 1 (2:2) = 0:0139 (x): prob. standard normal distr. P(X< x) 28

29 Conditional Probability Suppose r.v. X and Y have joint pmf p(x,y) p(1,1)=0.5, p(1,2)=0.1, p(2,1)=0.1, p(2,2)=0.3 Q: Calculate the pmf of X given that Y=1 p Y (1)=p(1,1)+p(2,1)=0.6 X sample space {1,2} p X Y (1 1) =P(X=1 Y=1) = P(X=1, Y=1)/P(Y=1) = p(1,1)/p Y (1) = 5/6 Similarly, p X Y (2,1) = 1/6 29

30 Expectation by Conditioning r.v. X and Y. then E[X Y] is also a r.v. Formula: E[X]=E[E[X Y]] Make it clearer, E X [X]= E Y [ E X [X Y] ] It corresponds to the law of total probability E X [X]= E X [X Y=y] P(Y=y) Used in the same situation where you use the law of total probability 30

31 Example r.v. X and N, independent Y=X 1 +X 2 + +X N Q: compute E[Y]? 31

32 Example 1 A company s network has a design problem on its routing algorithm for its core router. For a given packet, it forwards correctly with prob. 1/3 where the packet takes 2 seconds to reach the target; forwards it to a wrong path with prob. 1/3, where the packet comes back after 3 seconds; forwards it to another wrong with prob. 1/3, where the packet comes back after 5 seconds. Q: What is the expected time delay for the packet reach the target? Memoryless Expectation by condition 32

33 Example 2 Suppose a spam filter gives each incoming an overall score. A higher score means the is more likely to be spam. By running the filter on training set of (known normal + known spam), we know that 80% of normal s have scores of 1.5 ± 0.4; 68% of spam s have scores of 4 ± 1. Assume the score of normal or spam follows normal distr. Q1: If we want spam detection rate of 95%, what threshold should we configure the filter? Q2: What is the false positive rate under this configuration? 33

34 Example 3 A ball is drawn from an bottle containing three white and two black balls. After each ball is drawn, it is then placed back. This goes on indefinitely. Q: What is the probability that among the first four drawn balls, exactly two are white? P (X = k) f(k; n; p) = Ã n k! p k (1 p) n k 34

35 Example 4 A type of battery has a lifetime with ¹=40 hours and ¾=20 hours. A battery is used until it fails, at which point it is replaced by a new one. Q: If we have 25 batteries, what s the probability that over 1100 hours of use can be achieved? Approximate by central limit theorem 35

36 Example 5 If the prob. of a person suffer bad reaction from the injection of a given serum is 0.1%, determine the probability that out of 2000 individuals (a). exactly 3 (b). More than 2 individuals suffer a bad reaction? (c). If we inject one person per minute, what is the average time between two bad reaction injections? Poisson distribution (for rare event in a large number of independent event series) Can use Binomial, but too much computation Geometric 36

37 Example 6 A group of n camping people work on assembling their individual tent individually. The time for a person finishes is modeled by r.v. X. Q1: what is the PDF for the time when the first tent is ready? Q2: what is the PDF for the time when all tents are ready? Suppose X i are i.i.d., i=1, 2,, n Q: compute PDF of r.v. Y and Z where Y= max(x 1, X 2,, X n ) Z= min(x 1, X 2,, X n ) Y, Z can be used for modeling many phenomenon 37

38 Example 7 A coin having probability p of coming up heads is flipped until two of the most recent three flips are heads. Let N denote the number of heads. Find E[N] P(N=n) = P(Y 2 3,, Y n-1 3, Y n 2) 38